Second-order Guarantees of Gradient Algorithms over Networks

We consider distributed smooth nonconvex unconstrained optimization over networks, modeled as a connected graph. We examine the behavior of distributed gradient-based algorithms near strict saddle points. Specifically, we establish that (i) the renowned Distributed Gradient Descent (DGD) algorithm likely converges to a neighborhood of a Second-order Stationary (SoS) solution; and (ii) the more recent class of distributed algorithms, based on gradient tracking (termed SONATA), likely converges to exact SoS solutions, thus avoiding (strict) saddle points.